where the optimization variable is an absolutely continuous path \((x(t),m(t))\)  between \((x_0,m_0)\)  and \((x_1,m_1)\) .
The convexification of \(\tilde{c}\)  is \( c((x_0,m_0),(x_1,m_1))\min _{m_0^i,m_1^i} \tilde{c}((x_0,m_0^1),(x_1,m_1^1)) + \tilde{c}((x_0,m_0^2),(x_1,m_1^2))\)  where \(m_0^1 + m_0^2 = m_0\)  and \(m_1^1 + m_1^2 = m_1\) .
We now introduce a static formulation called semi-couplings introduced in [1]} and used in [2]} to derive an explicit algorithm.
Semi-couplings are couples of nonnegative Radon measures \(\gamma ,\pi \)  on \(X \times X\)  such that \(\gamma _1 = \alpha \)  and \(\pi _2 = \beta \) .
The static problem is the minimization over the set of semi-couplings of
\(C_K(\gamma ,\pi ) = \inf _{\gamma ,\pi }\int _{X \times X} c\left((x,\frac{\gamma }{\lambda }(x,y)),(y,\frac{\pi }{\lambda }(x,y))\right) d\lambda (x,y)\,,\) 
